Cones and ping-pong in three dimensions

Frieden, Gabriel; Gélinas, Félix; Soucy, Étienne

doi:10.2140/involve.2024.17.11

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Cones and ping-pong in three dimensions

Gabriel Frieden, Félix Gélinas and Étienne Soucy

Vol. 17 (2024), No. 1, 11–28

DOI: 10.2140/involve.2024.17.11

Abstract

We study the hypergeometric group in ${GL}_{3} (ℂ)$ with parameters $α = (\frac{1}{4}, \frac{1}{2}, \frac{3}{4})$ and $β = (0, 0, 0)$ . We give a new proof that this group is isomorphic to the free product $ℤ ∕ 4 ℤ * ℤ ∕ 2 ℤ$ by exhibiting a ping-pong table. Our table is determined by a simplicial cone in $ℝ^{3}$ , and we prove that this is the unique simplicial cone (up to sign) for which our construction produces a valid ping-pong table.

Keywords

hypergeometric group, free product of groups, ping-pong lemma

Mathematical Subject Classification

Primary: 20E06

Milestones

Received: 25 January 2022

Revised: 22 November 2022

Accepted: 17 December 2022

Published: 15 March 2024

Communicated by Jim Haglund

Authors

Gabriel Frieden
Université du Québec à Montréal Montréal, QC Canada

Félix Gélinas
Université du Québec à Montréal Montréal, QC Canada

Étienne Soucy
Université du Québec à Montréal Montréal, QC Canada

Vol. 17, No. 1, 2024